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Find all primes that can be represented by $n^3+1$, $n\in \Bbb N$.

This is a problem from a math olympiad contest, no answer provided.

For sure 2 is one of them (when $n=1$), are there others?


marked as duplicate by Dietrich Burde, user91500, Namaste, Siong Thye Goh, B. Goddard Aug 27 '17 at 18:29

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    $\begingroup$ A "duplicate" of this $\endgroup$ – Jyrki Lahtonen Aug 26 '17 at 13:35
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    $\begingroup$ Last comment is a duplicate of the first :). $\endgroup$ – bluemaster Aug 26 '17 at 14:40
  • $\begingroup$ For me, who asked the question, it is not the same question, as I'm not familiar with the notation $(n+1)|(n^3+1)$. $\endgroup$ – bluemaster Aug 26 '17 at 14:48
  • $\begingroup$ bluemaster, you should really learn what $m\mid n$ means, as it is essential for primes. $\endgroup$ – Dietrich Burde Aug 26 '17 at 16:17
  • $\begingroup$ I'm not sure about the benefits of learning $m|n$, perhaps you are right. But, the answer given below (+1), which is fully understandable by most people (even my 10 years old daughter who originally asked this question to me), possibly because only uses simple precalculus algebra factorization arguments and notations, didn't require $m|n$. $\endgroup$ – bluemaster Aug 26 '17 at 17:48

Notice that $$n^3+1=(n+1)(n^2-n+1)$$ And so, for $n^3+1$ to be prime, either $$n+1=1$$ or $$n^2-n+1=1$$ But keep in mind that $n=0$ does not work, since $1$ isn't prime.

Can you take it from here?


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